On the norms used in computing the structured singular value

Different norms are considered to replace the Euclidean norm in an algorithm given by M.H.K. Fan and A.L. Tits (ibid., vol.33, p.284-9, 1988), which is used for the computation of the structured singular value of any matrix. The algorithm is explained, and it is shown that the l₁ norm is the best possible norm in a certain sense     

Robust stability and structured uncertainty bounded by the Euclidean norm

In the context of robust stability, the μ‐problem is generalized for uncertainty bounded by means of the Euclidean norm. In some cases, a weighted Euclidean norm may be preferable to the infinite norm, for example, when the deviation from the nominal parameters exhibits a Gaussian distribution, also in the case that the parameters of the system are estimated by the ellipsoid algorithm. Several polynomial‐time upper bounds for the new μ‐problem are proposed. Copyright © 2001 John Wiley & Sons, Ltd.     

A novel margin-based twin support vector machine with unity norm hyperplanes

For classification problems, twin support vector machine (TWSVM) determines two nonparallel hyperplanes by solving two related SVM-type problems. TWSVM classifies binary patterns by the proximity of it to one of the two nonparallel hyperplanes. Thus, to calculate the distance of a pattern from the hyperplane, we need the unity norm of the normal vector of the hyperplane. But in the formulation of TWSVM, these equality constraints were not considered. In this paper, we consider unity norm constraints by using Euclidean norm and add a regularization term with the idea of maximizing some margin in TWSVM and propose a novel margin-based twin support vector machines with unity norm hyperplanes (UNH-MTSVM). We solved UNH-MTSVM by Newton’s method, and the solution is updated by conjugate gradient method. The performance of both the linear and nonlinear UNH-MTSVM is verified experimentally on several bench mark and synthetic datasets. Experimental results show the effectiveness of our methods in both computation time and classification accuracy.     

Edge-backpropagation for noisy logo recognition

In this paper, we propose a new approach to improve the performance of multilayer perceptrons operating as autoassociators to classify graphical items in presence of spot noise on the image. The improvement is obtained by introducing a weighed norm instead of using the Euclidean norm to measure the input-output accuracy of the neural network. The weights used in the computation depend on the gradient of the image so as to give less importance to uniform colour regions, like the spots. A modified learning algorithm (edge-backpropagation) is derived from the classical backpropagation by considering the new weighed error function. We report a set of experimental results on a database of 134 company logos corrupted by artificial noise which show the effectiveness of the proposed approach.     

Soft learning vector quantization and clustering algorithms based on non-Euclidean norms: single-norm algorithms

This paper presents the development of soft clustering and learning vector quantization (LVQ) algorithms that rely on a weighted norm to measure the distance between the feature vectors and their prototypes. The development of LVQ and clustering algorithms is based on the minimization of a reformulation function under the constraint that the generalized mean of the norm weights be constant. According to the proposed formulation, the norm weights can be computed from the data in an iterative fashion together with the prototypes. An error analysis provides some guidelines for selecting the parameter involved in the definition of the generalized mean in terms of the feature variances. The algorithms produced from this formulation are easy to implement and they are almost as fast as clustering algorithms relying on the Euclidean norm. An experimental evaluation on four data sets indicates that the proposed algorithms outperform consistently clustering algorithms relying on the Euclidean norm and they are strong competitors to non-Euclidean algorithms which are computationally more demanding.     

基于时间序列的感知QoS的云服务组合

研究基于时间序列的感知QoS的云服务组合,将服务的QoS偏好随时间不断变化的过程纳入云服务组合的研究范围,将云服务组合建模成时间序列的相似度对比问题。分别用欧几里得距离和扩展Frobenius范数距离度量二维时间序列的相似度,继而用基于主成分分析的扩展Frobenius范数距离和欧几里得距离、Brute Force等方法度量多维时间序列的相似度,通过实验对比验证扩展Frobenius范数距离度量相似度在时间和准确性上的优越性。关     

Optimal left and right additive Schwarz preconditioning for minimal residual methods with Euclidean and energy norms

For the solution of non-symmetric or indefinite linear systems arising from discretizations of elliptic problems, two-level additive Schwarz preconditioners are known to be optimal in the sense that convergence bounds for the preconditioned problem are independent of the mesh and the number of subdomains. These bounds are based on some kind of energy norm. However, in practice, iterative methods which minimize the Euclidean norm of the residual are used, despite the fact that the usual bounds are non-optimal, i.e., the quantities appearing in the bounds may depend on the mesh size; see [X.-C. Cai, J. Zou, Some observations on the l² convergence of the additive Schwarz preconditioned GMRES method, Numer. Linear Algebra Appl. 9 (2002) 379-397]. In this paper, iterative methods are presented which minimize the same energy norm in which the optimal Schwarz bounds are derived, thus maintaining the Schwarz optimality. As a consequence, bounds for the Euclidean norm minimization are also derived, thus providing a theoretical justification for the practical use of Euclidean norm minimization methods preconditioned with additive Schwarz. Both left and right preconditioners are considered, and relations between them are derived. Numerical experiments illustrate the theoretical developments.     

基于核函数距离测度的加权模糊C均值聚类与Markov空域约束的快速鲁棒图像分割

由于模糊C均值聚类算法（FCM）存在两大缺陷：（1）对于非球形的数据聚类形式鲁棒性不够；（2）只考虑图像中的数值特征信息，而忽略了像素间的空间约束关系，因此FCM算法在对含有噪声的图像进行分割时缺乏足够的鲁棒性。针对以上问题，本文提出了一种结合Markov空域约束与基于核函数距离测度的加权模糊c均值聚类的快速鲁棒图像分割方法。为克服缺点（1），我们使用基于核函数的距离测度取代FCM中的欧氏距离，并使用加权模糊聚类的方式保证了计算的简洁性。与此同时，我们用Markov随机场描述图像的空域约束信息，并且通过数据融合的方法将模糊分割结果与空域约束信息结合在一起，从而得到既包含像素数值特征又包括空域约束信息的图像分割场。这样既克服了传统模糊C均值聚类算法的缺点，又最大限度地保证了分割算法计算的简单有效性。     

State-space measures for stability robustness

Stability robustness measures for a perturbed linear feedback system are derived based on state-space models of the system. The system may be a continuous-time or discrete-time system. The perturbations are modeled as additive perturbation matrices. Necessary and sufficient conditions for the stability of the perturbed closed-loop system for all perturbations of norm bounded by some positive number are obtained. The destabilizing perturbations of minimal norm are characterized. It is shown by an example that there are cases when the destabilizing perturbations of minimal norm are all complex. The results are expressed in terms of induced operator norms. These are later specialized to the Euclidean norm and expressed in terms of singular values. A simple example is also included to illustrate an application of the results of this note.     

基于同态加密算法的欧氏距离外包计算协议

针对外包存储数据在密文状态下有关欧氏距离无法计算的问题,构建了欧氏距离外包计算协议,降低了用户的计算负担,保护了数据隐私。回顾了分布式双陷门公钥密码方案。基于同态加密算法设计了安全的乘法协议、单个密钥加密下的完全平方式协议和联合公钥加密下的完全平方式协议,基于这三个基础计算协议设计了欧氏距离的外包计算协议。安全性分析表明该协议足够安全,效率分析显示该协议较为高效,并较好地解决了有关欧氏距离的外包计算问题,对于图像处理的发展有一定的促进作用。     

A gradient method for computing the optimal H/sub infinity / norm

A fast algorithm for computing the optimal H/sub infinity / norm is presented. The algorithm is based on certain explicit computations of gradients. Examples are given to show that the gradient method is much faster than the bisection method. The gradient method can be used without the upper and lower bound computations. This could be a substantial additional saving because computing these bounds requires the computation of the H/sub infinity / norm.<>     

Parallel computation of the Euclidean distance transform on a three-dimensional image array

In a two- or three-dimensional image array, the computation of Euclidean distance transform (EDT) is an important task. With the increasing application of 3D voxel images, it is useful to consider the distance transform of a 3D digital image array. Because the EDT computation is a global operation, it is prohibitively time consuming when performing the EDT for image processing. In order to provide the efficient transform computations, parallelism is employed. We first derive several important geometry relations and properties among parallel planes. We then, develop a parallel algorithm for the three-dimensional Euclidean distance transform (3D-EDT) on the EREW PRAM computation model. The time complexity of our parallel algorithm is O(log/sup 2/ N) for an N/spl times/N/spl times/N image array and this is currently the best known result. A generalized parallel algorithm for the 3D-EDT is also proposed. We implement the proposed algorithms sequentially, the performance of which exceeds the existing algorithms (proposed by Yamada, 1984). Finally, we develop the corresponding parallel programs on both the emulated EREW PRAM model computer and the IBM SP2 to verify the speed-up properties of the proposed algorithms.     

Partial condition number for the equality constrained linear least squares problem

In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the data space and the solution space are measured by the weighted Frobenius norm and the Euclidean norm, respectively. Then, we investigate the corresponding structured partial condition number when the problem is structured. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and two algorithms are devised. The obtained results are illustrated by numerical examples.     

Close tracking of equilibrium paths

A method to control a generic system of nonlinear ordinary differential equations between equilibrium states is analyzed. The objective is to ensure that the system's state space trajectory closely tracks an equilibrium path. The control law is obtained via time parameterization of the corresponding equilibrium control path. Conditions which guarantee that the system's state space trajectory closely tracks the equilibrium path are proved using two approaches. One approach uses the mean value theorem, and the other uses the slowly time‐varying systems theory. Importantly, both methods provide relationships between the control rate norm and the tracking error norm. These allow computation of upper bounds on the control rate norm which guarantee a desired upper bound on the tracking error norm. They also enable computation of upper bounds on the tracking error norm for a given upper bound on the control rate norm. Examples illustrate the theoretical results.     

Convolution and Hankel operator norms for linear systems

Some norms are derived for convolution and Hankel operators associated with linear, time-invariant systems. In certain cases, these norms are shown to be identical. The tightest possible bound has been obtained for the absolute magnitude of the Euclidean 2 or ∞ norm of the time-domain response of a multioutput system to certain classes of input disturbance     

Abstract geometrical computation 3: black holes for classical and analog computing

The so-called Black Hole model of computation involves a non Euclidean space-time where one device is infinitely “accelerated” on one world-line but can send some limited information to an observer working at “normal pace”. The key stone is that after a finite duration, the observer has received the information or knows that no information was ever sent by the device which had an infinite time to complete its computation. This allows to decide semi-decidable problems and clearly falls out of classical computability. A setting in a continuous Euclidean space-time that mimics this is presented. Not only is Zeno effect possible but it is used to unleash the black hole power. Both discrete (classical) computation and analog computation (in the understanding of Blum, Shub and Smale) are considered. Moreover, using nested singularities (which are built), it is shown how to decide higher levels of the corresponding arithmetical hierarchies.     

基于量子行为的微粒群优化算法的数据聚类*

在PSO聚类算法的基础上,提出了基于量子行为的微粒群优化算法(QPSO)的数据聚类.QPSO算法不仅参数个数少、随机性强,并且能覆盖所有解空间,保证算法的全局收敛.PSO与QPSO算法的不同在于聚类中心的进化上,实验中用到四个数据集比较的结果,证明了QPSO优于PSO聚类方法.在聚类过程中使用了一种新的度量代替Euclidean标准,实验证明了新的度量方法比Euclidean标准更具有健壮性,聚类的结果更精确.     

Infinite-dimensional convex optimization in optimal and robustcontrol theory

Many engineering problems can be shown to be equivalent to solving semidefinite programs (SPs), i.e., convex optimization problems involving linear matrix inequalities (LMIs). Powerful computation tools are available for such problems in the finite-dimensional case. However, the problems arising in optimal and robust control theory are often infinite dimensional, and so adequate computation tools are not available. The key to tackling such problems with finite computation tools is to have a primal-dual formulation of the problem without duality gap. In this paper we study infinite-dimensional SPs and present a lifting technique to recast SPs as parameterized linear programs (LPs). This enables the wealth of theoretical tools available for infinite-dimensional LPs to be extended to infinite-dimensional SPs. In particular, we develop some new sufficient conditions for the lack of a duality gap for infinite-dimensional SPs and give an exact characterization of the primal and dual problems for these cases. Both primal and dual problems are formed as infinite-dimensional SP problems, with finite truncations to each giving upper and lower bounds, respectively, on the exact solution to the infinite-dimensional problem. Thus, these results can form the basis of practical computation schemes for infinite-dimensional problems, which require only finite-dimensional computation tools. To illustrate the power of these tools we apply the results to two previously unsolved optimization problems, namely minimizing the l₁ norm of a closed-loop system subject to bounds on the frequency response magnitude at a finite number of points and/or bounds on the H₂ norm     

A simple procedure for the exact stability robustness computation of polynomials with affine coefficient perturbations

This paper considers the problem of the stability robustness computation of polynomials with coefficients which are affine functions of the parameter perturbations. A polynomial is said to be stable if its roots are contained in an arbitrarily pre-specified open set in the complex plane, and its stability robustness is then measured by the norm of the smallest parameter perturbation which destabilizes the polynomial. A simple and numerically effective procedure, which is based on the Hahn-Banach theorem of convex analysis and which is applicable for any arbitrary norm, is obtained to compute the stability robustness. The computation is then further simplified for the case when the norm used is the Hölder ∞-norm, 2-norm or 1-norm.